You are correct that $\partial u/\partial x_i$ is not a vector field on $N$. It is a section of $u^*TN$, i.e. associating to each point $x \in M$ a vector $\partial u/\partial x_i \in T_{u(x)} N$. So we take the second fundamental form $A$ of $N$, which is a section of $S^2 T^*N \otimes T^{\perp} N$, and we pull it back to form $u^*A$, which you denote $A(u)$, a section of $S^2 u^* T^*N \otimes u^* T^{\perp} N$. Then we plug in two sections $\partial u/\partial x_i, \partial u/\partial x_j$ of $u^* TN$.

You are correct that $\partial u/\partial x_i$ is not a vector field on $N$. It is a section of $u^*TN$, i.e. associating to each point $x \in M$ a vector $\partial u/\partial x_i \in T_{u(x)} N$. So we take the second fundamental form $A$ of $N$, which is a section of $S^2(T^*N) \otimes T^{\perp} N$, and we pull it back to form $u^*A$, which you denote $A(u)$, a section of $S^2(u^* T^*N) \otimes u^* T^{\perp} N$. Then we plug in two sections $\partial u/\partial x_i, \partial u/\partial x_j$ of $u^* TN$.